Does the Moon Orbit the Earth, or the Sun?

The Moon that went up a Hill but came down a planet over at Bad Astronomy

[Y]ou have to look at something called the Hill sphere. Basically, it’s the volume of space around an object where the gravity of that object dominates over the gravity of a more massive but distant object around which the first object orbits.
OK, in English — and more pertinent to this issue — it’s the volume of space around the Earth where the Earth’s gravity is more important than the Sun’s. If something is orbiting the Earth inside Earth’s Hill’s sphere, it’ll be a satellite of the Earth and not the Sun.